MEN" OF MATHEMATICS point of rz'exe behind this ingenious recasting of the algebra of complex numbers which are of importance for the sequel. If as usual i denotes V — 1, a 'complex number' is a number of the type a -f bi, where a,b are 'real numbers' or, if preferred, and more generally, elements of the field F defined by the above postulates. Instead of regarding a + & as one ^tunber5, Hamilton conceived it as an ordered couple of 'numbers', and he designated this couple by writing it (a,b). He then proceeded to impose definitions of sum and product on these couples, as suggested by the formal rules of combination sublimated from the experience of algebraists in manipulating complex numbers as if the laws of common algebra did in fact hold for them. One advantage of this new way of approaching complex numbers was this: the definitions for sum and product of couples were seen to be instances of the general, abstract definitions of sum and product as in a field. Hence, if the con- sistency of the system defined by the postulates for a field is 'proved, the like follows, without further proof, for complex numbers and the usual rules by which they are combined. It will be sufficient to state the definitions of sum and product in Hamilton's theory of complex numbers considered as couples («,&} (c4), etc. The sum of (aj>) and (c,rf) is (a -f b, c + d); their product is (ac — bd, ad ~- be). In the last, the minus sign is a's in a field; namely, the element on postulated in IV is denoted by — a. To the 0,1 of a field correspond here the couples (0,0), (1,0). With these definitions it is easily verified that Hamilton's couples satisfy all the stated postulates for a field. But they also accord with the formal rules for manipulating complex numbers. Thus, to (a.b). ic.d) correspond respectively a -f bi, c -j- di, and the formal "sum" of these two is (a. -f c) -f i(b + <9» to ^bieh corre- sponds the couple (a -f c, & -f d). Again, formal multiplication of a -f bi, c -r- id gives (ac - bd) -f i(ad -f- be), to which cor- responds the couple (ac - bd, ad + be). If this sort of thing is new to any reader, it will repay a second inspection, as it is an example of the way in which modem mathematics eliminates mystery. So long as there is a shred of mystery attached to any concept that concept is not mathematical. 392